week10 3d transformations nov 29 final.ppt

7/21/2019 week10 3d transformations nov 29 final.ppt

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PE-3011 :CAD/FEA

Prof. CHARLTON S. INAODefence University College of Engineering

3D

Transformations



Instructional Objectives

At the end of the session, the

students should be able to

demonstrate knowledge and be able

to:• Discuss the 3D transformation

concepts

• Discuss the algorithms for removing

hidden line, shading and coloring



Background/Review



3D Transformations• 2D transor!ation can "e e#tended to 3D transor!ation

in 3D s$ace using %o!ogenous coordinate s&ste! wit'!atri# !et'od(

)# & *+ )#(' &(' *(' '+

• ,ransation:

• .caing:

,

1000

000

000

000

z

y

x

S

S

S

1

0100

0010

0001

z y x T T T



Reection!irroring:

−

−

=

−

−=

−

−=

−

−−=

−

=

−=

−=

1000

0100

0010

0001

1000

0100

00100001

1000

0100

00100001

1000

0100

00100001

:Re

1000

0100

0010

0001

1000

0100

0010

0001

1000

0100

0010

0001

:Re

z

x xOrigin

yz xz xy

T

T T T

axisanabout flection

T T T

planeaabout flection



• Rotation: n 3D we rotation is a"out an a#is nota"out a $oint ike in 2D and $ositive ange is taken to"e ccw rotation(

Rotation about Z-axis Rotation about X-axis

Rotation about Y-axis

−=

1000

0100

00

00

θ θ

θ θ

CosSin

SinCos

R Z

−=

1000

00

00

0001

β α

α α

CosSin

SinCos R

X

−

=

1000

00

0010

00

φ φ

φ φ

CosSin

SinCos

RY





• Example: Rotation o a $oint P# & * a"out an& ar"itrar& a#is$assing t'roug' P1d e and P2g ' i( ,'e direction or t'eunit a#is vector u is deter!ined "& t'e s$eci4ed rotationdirection :

),,(|21|

21 cba P P P P u =

−−=











−=

1000

0//0

0//0

0001

)(d cd b

d bd c R

X α

−=

1000

00

00

0001

β α

α α

CosSin

SinCos R

X



( )

−=

1000

00

0010

00

d a

ad

RY β

−

=

1000

00

0010

00

β β

β β

CosSin

SinCos

RY



.te$ 3: Peror! t'e re5uired transor!ation rotationa"out t'e *- a#is(

−

=

1000

0100

00

00

)(θ θ

θ θ

θ CosSin

SinCos

R Z



*Go away from the origin,

back to initial position



EMOVING HIDDEN LINES

AND

SURFACES ALGORITHMS



REMOVI! "IDDE #IE$ %D $&R'%(E$ %D%#O!RIT"M$

Edges and suraces o a given o"6ect !a&

"e 'idden or visi"e in a given i!agede$ending on t'e viewing Direction(

,'e deter!ination o 'idden edges and

suraces can "e considered as a visi"iit&$ro"e!(

,'e soution to t'e $ro"e! o re!oving

'idden edges and surace draws onvarious conce$ts ro! co!$uting !ain&sorting and geo!etric !odeing !ain&$ro6ection and intersection(



A wide variet& o 'idden ine and 'iddensurace re!oving agorit'! is ine#istencetoda&( ,'e t&$e o gra$'ics dis$a& devices t'e&

su$$ort and "& t'e t&$e o data structureor geo!etric !odeing t'e& o$erate on-

wire ra!e surace or soid !odeinginuence t'e deveo$!ent o t'eseagorit'!s(.o!e s&ste!s use $arae $rocessing and

so!e use seria $rocessing( ,'e avaia"e agorit'!s can "e cassi4ed

into 'idden ine 'idden surace and 'iddensoid re!ova agorit'!s(



%idden ine and 'idden surace agorit'!s'ave "een cassi4ed as:

)* Ob+et $pae meto.s

/* Ima0e $pae meto.s

3* "1bri. Meto.s 2i is aco!"ination o

"ot'(An o"6ect 7s$ace agorit'! utii*e t'e s$atia

and geo!etrica reations'i$s a!ong t'eo"6ect in t'e scene to deter!ine 'idden and

visi"e $art o t'ese o"6ects(An i!age agorit'! concentrates on t'e

4na i!age to deter!ine w'at is visi"ewit'in eac' raster $i#e in t'e case o raster

dis$a&s(



2(0 !age s$ace !et'ods can "e urt'ercassi4ed as

/) Raster %l0oritms: ,'e& use a $i#e!atri#

re$resentation o t'e i!age(// Vetor %l0oritms: ,'e& use end $oint

coordinates o t'e ineseg!ent orre$resenting t'e i!age(

9ost 'idden surace agorit'!s use rasteri!age s$ace !et'ods w'ie !ost 'idden ine

agorit'! use o"6ect-s$ace !et'ods( ,'e o"6ect s$ace agorit'! are !ore

accurate t'an i!age s$ace agorit'!(



Re4ie2 56a70roun.

• - .troke writing and

• - Raster .can

• $tro7e 2ritin05 line.ra2in05ran.om position: eectron"ea! is directed ro! one $oint tot'e ot'er ike a $enci to create a ineand an i!age is or!ed out o t'ese5uence o straig't ine seg!ents(



Vetor or $tro7e 2ritin0



Raster $an:

• ,'e viewing screen is divided into arge nu!"er odiscrete $'os$'or $icture ee!ents caed Pi#es(

• ,'e !atri# o $i#e constitutes t'e raster(eg(2;#2;102<#102<

• Eac' $i#e can "e !ade to gow wit' a di=erent"rig'tness and coors(• An eectron "ea! creates t'e i!age "& swee$ing

aong a 'ori*onta ine ro! et to rig't andenergi*ing t'e $i#es in t'at ine and !ove to t'ene#t ine "eow and $roceeds in a 4#ed $attern asin 4g(

• ,'e $rocess is re$eated at rate o 30-;0 entirescans $er second



Pi#e



%l0oritm

• For a given t'ree di!ensiona scene a given viewing $oint anda given direction ei!inate ro! an a$$ro$riate two-di!ensiona

$ro6ection o t'e scene a $arts o edges and aces w'ic' t'eo"server cannot see(

• For ort'ogra$'ic $ro6ection t'e ocation o t'e viewing $oint isnot needed( ,'e two di!ensiona $ro6ection is aso known as t'ei!age $icture s$ace(



• A soid !ode is an idea

re$resentation( ,'e root o t'e treere$resents t'e t'ree di!ensiona scenei(e( Data"ases o wire ra!es andsuraces $o&gons and t'e order ot'eir edges cockwise(

• ,'ese !odi4cations are t&$ica&ac'ieved in an interactive wa& "&

re5uesting t'e user to digiti*e t'eedges o eac' ace in a given order(



• ,'e second ste$ is to a$$& t'e

$ro$er geo!etric transor!ations"ased on t'e viewing direction to t'et'ree-di!ensiona scene data too"tain t'e two di!ensiona i!age

>raw? data(• ,'ese transor!ations are used to

$roduce t'e de$t' inor!ation w'ic'is stored in t'e i!age data"ase or

de$t' co!$arison $ur$ose(



• ,'e ne#t ste$ are sorting and a$$&ingvisi"iit& tec'ni5ues deat esew'ere(@it' t'e co!$etion o t'e sorting

according to t'e visi"iit& criteria set"& t'e visi"iit& tec'ni5ues t'e'idden edges are identi4ed and

re!oved ro! t'e i!age data( ,'east ste$ in t'e agorit'! is to dis$a&or $ot t'e 4na i!ages(



SHADING

AND

COLOURING



$"%DI! %D (O#O&RI!.'aded coor i!age conve& s'a$e

inor!ation t'at cannot "ere$resented in ine drawings(

.'aded i!ages can aso conve&

eatures ot'er t'an s'a$e suc' assurace 4nis' or !ateria t&$e$astic or !etaic ook s'aded

i!age rendering agorit'!s 4terinor!ation "& dis$a&ing on& t'evisi"e surace(



9an& s$atia reations'i$s t'at are unresoved insi!$e wire ra!e dis$a&s "eco!e cear wit'

s'aded dis$a&s(Critica s'ading scene a $in'oe ca!era !ode is

a!ost universa& used(Rendering "egins "& soving t'e 'idden surace

re!ova $ro"e! to deter!ine w'ic' o"6ects

and/or $ortions o o"6ects are visi"e in t'e scene(As t'e visi"e surace are ound t'e& !ust "e

"roken down into $i#es and s'aded correct&( ,'is $rocess !ust take into account t'e $osition

and coor o t'e ig't sources and t'e $ositionand coor o t'e ig't sources and t'e $ositionorientation and surace $ro$erties o t'e visi"eo"6ects(



• ,'e various s'ading agorit'!s areavaia"e( A!ong t'e!

1( constant s'ading

2( ourad or 4rst derivative s'ading and

3( P'ong or second derivative s'ading

agorit'!s are ver& $o$uar(

• Coors aciitate creating geo!etr& anddis$a& i!age( Coors can "e used in

geo!etric construction( n t'is casevarious wire ra!e surace or soidentities can "e assigned di=erent coors todistinguis' t'e!(



Coor is one o t'e two !ainingredients t'e second "eing te#tureo s'aded i!ages $roduced "&

s'ading agorit'!s in so!eengineering a$$ications suc' as4nite ee!ent ana&sis coors can "e

e=ective& to dis$a& contour i!agessuc' as stress or 'eat-u# contours(

Two Main Ingredients:

1) Color

2) Texture



9icro$rocessor %eat .ink



E#a!$e o stress and 'eat u#contour o $iston



.tructura Ana&sis o 9ug@'ee



,ransonic Fow Around a 2Dairoi

i d



,ransonic Fow Around a 2Dairoi



eiet .u$er 9ac' Ra*or



3D si!uation on a ,unne



Constant .'ading• .'ading /Constant .'ading is a $rocess used in

drawing or de$icting eves o darkness on $a$er"& a$$&ing !edia !ore dense& or wit' a darkers'ade or darker areas and ess dense& or wit' aig'ter s'ade or ig'ter areas( ,'ere are various

tec'ni5ues o s'ading incuding cross 'atc'ingw'ere $er$endicuar ines o var&ing cosenessare drawn in a grid $attern to s'ade an area( ,'ecoser t'e ines are toget'er t'e darker t'e areaa$$ears( ikewise t'e art'er a$art t'e ines aret'e ig'ter t'e area a$$ears(

• ig't $atterns suc' as o"6ects 'aving ig't ands'aded areas 'e$ w'en creating t'e iusion ode$t' on $a$er(



1)Rendered image of a box. This image has

no shading on its faes! but uses edge lines

to se"arate the faes.

2)This is the same image with the edge lines

remo#ed.

$)This is the same image rendered with

shading of the faes to alter the olors of the

$ faes based on their angle to the lightsoures



Constant .'ading



ourad .'ading!ourau. sa.in0 na!ed ater %enri

ouraud is an inter$oation !et'od usedin co!$uter gra$'ics to$roduce continuous s'ading o suracesre$resented "& $o&gon !es'es( n$ractice ouraud s'ading is !ost otenused to ac'ieve continuous ig'ting ontriange suraces "& co!$uting t'e ig'ting

at t'e corners o eac' triange and inear&inter$oating t'e resuting coours oreac' $i#e covered "& t'e triange(ouraud 4rst $u"is'ed t'e tec'ni5ue in

11(



• ouraud s'ading is considered su$eriorto at s'ading w'ic' re5uiressigni4cant& ess $rocessing t'an P'ongs'ading "ut usua& resuts in a acetedook(

• n co!$arison to P'ong s'ading ouraud

s'adings strengt' and weakness ies in itsinter$oation( a !es' covers !ore $i#esin screen s$ace t'an it 'as verticesinter$oating coour vaues ro! sa!$es

o e#$ensive ig'ting cacuations atvertices is ess $rocessor intensive t'an$eror!ing t'e ig'ting cacuation oreac' $i#e as in P'ong s'ading(



• %owever 'ig'& ocai*ed ig'ting e=ects suc'as s$ecuar 'ig'ig'ts e(g( t'e gint o reectedig't on t'e surace o an a$$e wi not "e

rendered correct& and i a 'ig'ig't ies in t'e!idde o a $o&gon "ut does not s$read to t'e$o&gons verte# it wi not "e a$$arent in aouraud rendering

• Converse& i a 'ig'ig't occurs at t'e verte# o a$o&gon it wi "e rendered correct& at t'is verte#as t'is is w'ere t'e ig'ting !ode is a$$ied "utwi "e s$read unnatura& across a neig'"oring$o&gons via t'e inter$oation !et'od(



ourad .'ading



P'ong• P'ong s'ading reers to an inter$oation

tec'ni5ue or surace s'ading in 3Dco!$uter gra$'ics( t is aso caedP'ong inter$oation)1+ or nor!a-vector

inter$oation s'ading()2+ .$eci4ca& itinter$oates surace nor!as acrossrasteri*ed $o&gons and co!$utes $i#ecoors "ased on t'e inter$oatednor!as and a reection !ode( P'ongs'ading !a& aso reer to t'e s$eci4cco!"ination o P'ong inter$oation and

t'e P'ong reection !ode(

P'



P'ong



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week10 3d transformations nov 29 final.ppt